How to make the shapekey influence separate object's position?
You could use vertex parenting to have the objects follow parts of your mesh. This has similarities to the hook modifier.

Select the teeth, shift select the head, go into edit mode, select the target vertex (only one or three vertices can be selected, three vertices allows rotation when following) press CtrlP to parent the other selected object to the selected vertex.



Be careful as I expect you will deform the face for animation and the parenting will keep working. You could add an extra unattached vertex that moves with the neck shapekey but isn't used in facial animation, or maybe you could choose three vertices located one in each ear and one at the top of the head that won't get deformed in animation. You may also need the armature to follow so that any animation will happen relative to the new location, or parent a facial armature which will take the teeth and eyes with it.